Subgroup ($H$) information
| Description: | not computed |
| Order: | \(23276\)\(\medspace = 2^{2} \cdot 11 \cdot 23^{2} \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | not computed |
| Generators: |
$c^{22}d^{8}, d, b^{2}c^{308}d^{3}, b^{11}c^{198}d^{9}, ab^{8}c^{236}d^{5}$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, solvable, and an A-group. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $F_{23}\wr C_2$ |
| Order: | \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| Exponent: | \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $F_{23}\wr C_2$ |